On the semigroup algebra of binary relations∗

The semigroup of binary relations on {1, . . . , n} with the relative product is isomorphic to the semigroup Bn of n×n matrices over {0, 1} with the Boolean matrix product. Over any field F , we prove that there is an ideal Kn in FBn of dimension (2n−1)2, and we construct an explicit isomorphism of Kn with the matrix algebra M2n−1(F ). Let Zn = {1, . . . , n} (n ≥ 1); then P (Z n), the power set of the ordered pairs, is the set of binary relations on Zn. For X, Y ∈ P (Z n) the relative product is XY = { (i, k) | ∃j ∈ Zn with (i, j) ∈ X and (j, k) ∈ Y }. We identify X ∈ P (Z n) with the Boolean matrix X = (xij) where xij = 1 if (i, j) ∈ X and xij = 0 if (i, j) / ∈ X; the relative product coincides with the Boolean matrix product. Definition 1. The set Bn of all X = (xij) with the Boolean matrix product is the semigroup of Boolean matrices. Over any field F , the vector space FBn with the bilinear extension of the Boolean matrix product is the semigroup algebra of Boolean matrices. Clearly |Bn| = 2n2 . The subset of permutation matrices in Bn is a subgroup isomorphic to the symmetric group Sn. For the theory of Bn as an abstract semigroup see Schwarz [5]. Recent monographs on related topics are Maddux [4] and Jespers and Okniński [2]. Kim and Roush [3] studied linear representations of Bn, and showed that the ideal Kn (Definition 7 below) is isomorphic to the matrix algebra M2n−1(F ) (Corollary 21 below) ∗2000 MSC: Primary 20M25; Secondary 03G15, 16S36, 20M30. †Corresponding author. Email: [email protected]